New lower bound for the optimal congruent geodesic ball packing density of screw motion groups in H²\!×\!R space

In this paper, we present a new record for the densest geodesic congruent ball packing configurations in H²\!×\!R geometry, generated by screw motion groups. These groups are derived from the direct product of rotational groups on H² and some translation components on the real fibre direction R that can be determined by the corresponding Frobenius congruences. Moreover, we developed a procedure to determine the optimal radius for the densest geodesic ball packing configurations related to the considered screw motion groups. The highest packing density, ≈0.80529, is achieved by a multi-transitive case given by rotational parameters $(2,20,4)$. E. Moln\'{a}r demonstrated that homogeneous 3-spaces can be uniformly interpreted in the projective 3-sphere PS³(V⁴, V₄, R). We use this projective model of H²\!×\!R to compute and visualize the locally optimal geodesic ball arrangements.